Optimal. Leaf size=28 \[ \frac {x^5 \left (c^4+\frac {1}{x^4}\right )}{6 c^4 \sqrt {\text {sech}(2 \log (c x))}} \]
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Rubi [A] time = 0.04, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5551, 5549, 264} \[ \frac {x^5 \left (c^4+\frac {1}{x^4}\right )}{6 c^4 \sqrt {\text {sech}(2 \log (c x))}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 5549
Rule 5551
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {\text {sech}(2 \log (c x))}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{\sqrt {\text {sech}(2 \log (x))}} \, dx,x,c x\right )}{c^5}\\ &=\frac {\operatorname {Subst}\left (\int \sqrt {1+\frac {1}{x^4}} x^5 \, dx,x,c x\right )}{c^6 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=\frac {\left (c^4+\frac {1}{x^4}\right ) x^5}{6 c^4 \sqrt {\text {sech}(2 \log (c x))}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 44, normalized size = 1.57 \[ \frac {\left (c^4 x^4+1\right )^2 \sqrt {\frac {c^2 x^2}{2 c^4 x^4+2}}}{6 c^6 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 48, normalized size = 1.71 \[ \frac {\sqrt {2} {\left (c^{8} x^{8} + 2 \, c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{12 \, c^{6} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 39, normalized size = 1.39 \[ \frac {\sqrt {2}\, x \left (c^{4} x^{4}+1\right )}{12 \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}\, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 30, normalized size = 1.07 \[ \frac {{\left (\sqrt {2} c^{4} x^{4} + \sqrt {2}\right )} \sqrt {c^{4} x^{4} + 1}}{12 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 42, normalized size = 1.50 \[ \frac {{\left (c^4\,x^4+1\right )}^2\,\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4+1}}}{12\,c^6\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt {\operatorname {sech}{\left (2 \log {\left (c x \right )} \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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